Well, I would answer "no" as long as extra dimensions are not included in the theory. The unification of the Electromagnetic interactions with the Gravitational ones was attempted by people who tried to extend the dimensions to 5 at first. But this theory of extra dimensions has not been verified yet.
The thing is that the "electromagnetic" interactions happen within a different space than the known space-time- described by the spacetime-metric, the object that lets you measure distances on a (hyper)surface. As long as that space is not within the spacetime, it cannot affect the geometry of the last.

Well, I would answer "no" as long as extra dimensions are not included in the theory. The unification of the Electromagnetic interactions with the Gravitational ones was attempted by people who tried to extend the dimensions to 5 at first. But this theory of extra dimensions has not been verified yet.
The thing is that the "electromagnetic" interactions happen within a different space than the known space-time- described by the spacetime-metric, the object that lets you measure distances on a (hyper)surface. As long as that space is not within the spacetime, it cannot affect the geometry of the last.

This really isn't accurate.

It's not very difficult to unify electricity and magnetism with General Relativity. We don't know how to unify quantum electrodynamics with relativity, but classical E&M is pretty simple.

And yes, electromagnetic fields impact the curvature of space-time. But you need monstrous EM fields to do it by any appreciable amount. For an order-of-magnitude estimate, to get close to the same curvature that is caused by a mass ##m##, you need electromagnetic fields with energy around ##mc^2##.

And yes, electromagnetic fields impact the curvature of space-time. But you need monstrous EM fields to do it by any appreciable amount. For an order-of-magnitude estimate, to get close to the same curvature that is caused by a mass mm, you need electromagnetic fields with energy around mc2mc^2.

Well that is correct... I was thinking only of the first half of Einstein Equations and not the energy-momentum tensor.